[R] Linear optimization with quadratic constraints

Preetam Pal lordpreetam at gmail.com
Wed Jan 4 23:39:33 CET 2017

Hello guys,

The context is ordinary multivariate regression with k (>1) regressors,
i.e. *Y = XB + Error*, where
Y = n X 1 vector of predicted variable,
X = n X (k + 1) matrix of regressor variables(including ones in the first
B = (k+1) vector of coefficients, including intercept.

Say, I have already estimated B as B_hat = (X'X)^(-1) X'Y.

I have to solve the following program:

*minimize f(B) = LB*   ( L is a fixed vector 1 X (k+1)   )
such that:
*[(B-B_hat)' * X'X * (B-B_hat) ] / [ ( Y - XB_hat)' (Y - XB_hat) ] *  is
less than a given value *c*.

Note that this is a linear optimization program *with respect to B* with
quadratic constraints.

I don't understand how we can solve this optimization - I was going through
some online resources, each of which involve manually computing gradients
of the objective as well as constraint functions - which I want to avoid
(at least manually doing this).

Can you please help with solving this optimization problem? The inputs
would be:

   - X and Y
   - B_hat
   - L
   - c

Please let me know if any further information is required - the set-up is
pretty general.


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